Why is Psi^2 not square rooted in Schroedinger's Equation?

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Discussion Overview

The discussion centers around the interpretation of the wavefunction in Schrödinger's Equation, particularly why the square of the modulus of the wavefunction (Psi^2) is used to derive probability densities, rather than the wavefunction itself or its square root. Participants explore the mathematical and conceptual implications of using complex numbers in quantum mechanics and the historical context of these interpretations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that the complex plane allows for the representation of probability amplitudes on an imaginary axis, with the real axes reflecting spatial dimensions.
  • There is a proposal that the normalization process requires the integral of Psi^2 over all space to equal 1, leading to the interpretation of Psi^2 as a probability density.
  • Others argue that the integral of Psi itself does not yield a real-valued number or a direct physical interpretation, thus reinforcing the need for Psi^2.
  • A participant notes that Schrödinger originally viewed Psi^2 as representing charge density, while the probabilistic interpretation was later attributed to Max Born.
  • Some participants question the necessity of using complex numbers in Schrödinger's Equation, suggesting that a real variable could suffice for charge distribution.
  • Another participant emphasizes that the complex nature of the wavefunction is essential for deriving classical equations from wave equations, particularly in relation to dispersion relations.
  • Interference patterns in experiments, such as the double slit experiment, are cited as justification for the Born Rule, which states that probabilities are derived from the square of the modulus of the wavefunction.
  • Some participants express confusion over the historical development of the interpretation of Psi and its implications for quantum mechanics.

Areas of Agreement / Disagreement

Participants express a range of views on the interpretation of the wavefunction and the necessity of its complex nature. There is no consensus on the best interpretation or the implications of using complex numbers versus real variables.

Contextual Notes

Participants highlight the historical context of the development of quantum mechanics, particularly the differing interpretations of Psi by Schrödinger and Born. There are also discussions about the mathematical properties of the wavefunction and its implications for physical interpretations, which remain unresolved.

  • #31
unusualname said:
(because we actually are the quantum system we hope to influence)
If we (humans) also obey the laws of quantum mechanics, then we do not have free will. But the good news is that then we CAN use EPR correlations to send information faster than light:
http://xxx.lanl.gov/abs/1006.0338
 
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  • #32
Demystifier said:
If we (humans) also obey the laws of quantum mechanics, then we do not have free will. But the good news is that then we CAN use EPR correlations to send information faster than light:
http://xxx.lanl.gov/abs/1006.0338

As a consequence, by entangling a part of brain (responsible for the illusion of FW)

Ahhh, but the rub is, there would be no part of the brain responsible for the illusion of FW, it would only look that way to us. But still, that wouldn't alter your subjective illusion of being able to send information faster than light - live the dream.
 
  • #33
sujiwun said:
But still, that wouldn't alter your subjective illusion of being able to send information faster than light
Exactly! :approve:
 
  • #34
Demystifier said:
As long as Psi evolves deterministically and |Psi|^2 determines probability, we do NOT have the ability to influence randomness (i.e., no free will).

Why not? We might just have our choices limited by what Psi allows as possible states, but still be able to choose those states when we exercise free-will, at least in some limited form.

Evolution has had a few billion years to produce something very novel, eg a self-aware consciousness that is able to select the uncaused event coming out the other side of a Planckian black-hole singularity, or guide the microscopic dynamics at boundaries of complex higher dimensional structures where extra degrees of freedom suddenly become available (and hence we may have a one-to-many mapping which can't be deterministic)

The crucial point is that we cannot determistically influence any other quantum states except the ones we actually are made from.

In a sense all particles in the universe have "free-will" in this model, but only with a complex evolved consciousness are the choices made interesting beyond statistically random, and you can mostly describe the universe accurately with mathematical models such as thermodynamics. But no (generally applicable) deterministic model would predict Apollo 11 traveling to the Moon (for example)

That doesn't necessarily imply we are that special, just that we are doing something more interesting than non-conscious matter, and great works of art, philosophical and scientific thought, human emotions etc appears to our feeble sense as immensely deep and "meaningful".

But at least if we have randomness which we can partially influence we can truly say the future is unknown but the possibilities are (countably) infinite, which makes it worth getting out of bed in the morning :smile:
 
  • #35
unusualname said:
Why not? We might just have our choices limited by what Psi allows as possible states, but still be able to choose those states when we exercise free-will, at least in some limited form.

Evolution has had a few billion years to produce something very novel, eg a self-aware consciousness that is able to select the uncaused event coming out the other side of a Planckian black-hole singularity, or guide the microscopic dynamics at boundaries of complex higher dimensional structures where extra degrees of freedom suddenly become available (and hence we may have a one-to-many mapping which can't be deterministic)

The crucial point is that we cannot determistically influence any other quantum states except the ones we actually are made from.

In a sense all particles in the universe have "free-will" in this model, but only with a complex evolved consciousness are the choices made interesting beyond statistically random, and you can mostly describe the universe accurately with mathematical models such as thermodynamics. But no (generally applicable) deterministic model would predict Apollo 11 traveling to the Moon (for example)

That doesn't necessarily imply we are that special, just that we are doing something more interesting than non-conscious matter, and great works of art, philosophical and scientific thought, human emotions etc appears to our feeble sense as immensely deep and "meaningful".

But at least if we have randomness which we can partially influence we can truly say the future is unknown but the possibilities are (countably) infinite, which makes it worth getting out of bed in the morning :smile:


Doesn't QM make make evolution a mirage?
 
  • #36
sujiwun said:
Doesn't QM make make evolution a mirage?

eh?

Of course, the more sane veiw is that free-will is illusory, even if we have fundamental randomness in QM, since it seems more likely that evolution only works at molecular granularity and at best we may able to "reshape" the wave-function in some limited ways, but since that would have to obey Schrödinger evolution it would be a deterministic mechanism, even if the actual wave-function "collapse" is random. Then we're just like flotsam floating on an ocean, the future is undecided (no superdeterminism) but we can't really choose where it leads.
 
  • #37
unusualname said:
eh?

.

I wasn't suggesting that evolution doesn't exist, just that the order we see and call evolution is the product of randomness. That doesn't mean it is random - it still has the same structure we call evolution, the same order - just that its driving mechanism, is randomness, chaos.

In otherwords, "the fit survive" may be true in general - as part of the perceived order, nevertheless, just because you are fit it doesn't necessarily follow that you will - as part of the random chaos - you might still get hit by that bus tomorrow.

The philosophical implications as it relates to free will can be seen in history - fascism's "will to power" in a controlling attempt to create a race of ubermenschen vs humanistic liberalism.
 
  • #38
unusualname said:
Why not? We might just have our choices limited by what Psi allows as possible states, but still be able to choose those states when we exercise free-will, at least in some limited form.
Can a free will decide to end in the state |up> WHENEVER the wave function is |up>+|down>? If it can, then it contradicts quantum mechanics. If it cannot, then it is not free will. Does, either quantum mechanics is wrong or free will does not exist.
 
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  • #39
Demystifier said:
Can a free will decide to end in the state |up> WHENEVER the wave function is |up>+|down>? If it can, then it contradicts quantum mechanics. It it cannot, then it is not free will. Does, either quantum mechanics is wrong or free will does not exist.

Well, QM obviously isn't the full story, just a probabilistic mathematical model that works very well.

But eventually, physics will have to tackle the problem of human free-will and consciousness, since even String/M theory isn't a Theory of Everything (doesn't even attempt to solve the QM interpretation problem)

I think most physicists expect that consciousness & free-will is akin to a phase-change, a complex emergent phenomena from essentailly simple underlying physics, and free-will is itself not subverting deterministic Schrödinger Evolution, since there doesn't seem to be a workable explanation of how the molecular granularity of Evolution could create a physical mechanism that would work sub-quantum level.

But physics doesn't even tackle the underlying randomness of QM yet, there is no agreed understanding of why QM appears random. If there is an underlying mechanism producing uncaused events, then it gives some hope that free-will may be possible. But for uncaused (nondeterministic) events the microscopic dynamics either need a singularity or boundary points where degrees of freedom suddenly increase.

There is some speculation (and some papers on arXiv) that particles themselves may be microscopic black holes, but even then the cosmic censorship hypothesis suggests naked singularities cannot form.

So for now we are stuck with a determistically evolving Psi and a random "collapse" mechanism. But we have no idea why this is so. The dBB interpretation suggests that the probabilities are simply describing an underlying deterministic dynamics, and that is appealing since we have the well-known notion of invariant measures (probability densities) in ergodic theory which can be applied.

But that just seems a little too convenient, and still leaves the huge problem of explaining the physical origin of Psi, and its non-local mechanism.

Its amazing that after 80 years we haven't progressed much on these issues.
 
  • #40
unusualname said:
Well, QM obviously isn't the full story, just a probabilistic mathematical model that works very well.
I'm fine with that. But then deterministic Bohmian version of QM may also be compatible with free will, as long as you allow that Bohmian QM, even if more correct than standard QM, is not the final theory of everything either.

My point is that, concerning the issue of free will, probabilistic and deterministic interpretations of QM are equally good (or equally bed), so one should use free-will arguments to prefer one over the other.
 
  • #41
Demystifier said:
I'm fine with that. But then deterministic Bohmian version of QM may also be compatible with free will, as long as you allow that Bohmian QM, even if more correct than standard QM, is not the final theory of everything either.

My point is that, concerning the issue of free will, probabilistic and deterministic interpretations of QM are equally good (or equally bed), so one should use free-will arguments to prefer one over the other.

Ok, but for (true) free-will you need non-determinism at some level of nature, Bohmian Mechanics makes it difficult (for me) to see where that non-determism can arise, since then, even if our evolved consciousness is able to "load the dice" wrt the decoherence mechanism for quantum state selection, BM still says it's a deterministic mechanism. So free-will will then require something quite novel, and currently unknown to physics, which would be frustrating, since I want to blimin' know how this all works in my lifetime.:smile:
 
  • #42
unusualname said:
Ok, but for (true) free-will you need non-determinism at some level of nature, Bohmian Mechanics makes it difficult (for me) to see where that non-determism can arise, since then, even if our evolved consciousness is able to "load the dice" wrt the decoherence mechanism for quantum state selection, BM still says it's a deterministic mechanism. So free-will will then require something quite novel, and currently unknown to physics, which would be frustrating, since I want to blimin' know how this all works in my lifetime.:smile:
It is very easy to add a non-deterministic component to Bohmian mechanics. Just modify the Bohmian velocity formula, by addding one additional stochastic velocity term the statistical average of which vanishes.
 
  • #43
Demystifier said:
It is very easy to add a non-deterministic component to Bohmian mechanics. Just modify the Bohmian velocity formula, by addding one additional stochastic velocity term the statistical average of which vanishes.

I suppose so, but it would then require new ideas to explain that stochastic mechanism, which might be built on another (lower) level of deterministic dynamics and so on...

(The idea of deterministic chaotic dynamics generating the probability densities is appealing is some ways, but it still implies the future is already decided even if it is unknowable (not predictable). Physics needs an non-deterministic mechanism at some level, and I don't know why singularities (for example) are considered such a bad thing.)
 
  • #44
unusualname said:
Well, QM obviously isn't the full story, just a probabilistic mathematical model that works very well.

But eventually, physics will have to tackle the problem of human free-will and consciousness, since even String/M theory isn't a Theory of Everything (doesn't even attempt to solve the QM interpretation problem)

More likely that human psychology will explain physics than vice versa.
 

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